Parametric yield: Do you know what you miss?

PDC methods are still relatively cheap for certain technologies that have a limited number of technological parameters, but they could lead to an exponential explosion of possible corners for other technologies.

The main issues of PDC are:
1. Corners should be provided
2. Big number of corners with nanometer technologies
3. Possibility of over-design.
4. High risk of effective yield estimation
5. PDC is digital rules, each one indicates if corner passes or fails.

Other market tools and literature propose different approaches: classical Monte Carlo (MC), enhanced MC as importance sampling or worst-case analysis. The first one is the most accurate and guarantees a confidence interval on the estimation, but needs a huge number of runs. The more the designer is ready to invest in simulations the more the yield estimation will become accurate.

Enhanced MC or worse-case analysis needs fewer runs, to the detriment of accuracy.

The following fast study based on a 65nm LDO (Low-dropout regulator, circuit on Figure 1) is only to show how many MC runs could be needed to the stabilization of Monte Carlo computing.

Figure 2 shows the Phase-Margin statistical distributions for different number of MC runs and Figure 3 shows the mean, standard deviation, minimum and maximum values of these distributions.

Figure 1: LDO simplified schematics

These data show that for some critical lower and upper specification limits (near minimum and maximum of distribution values), at least some thousands Monte Carlo runs are required to provide a correct statistical distribution and a good yield accuracy and robustness! This result can be explained by the complex behavior of phase margin curve, and the difficulties to catch with Monte Carlo runs the phase margin optima (see Figure 5).